Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by
functions) and their rates of change in space and/or time (expressed as
derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various
forces acting on the body) to express these variables dynamically as a
differential equation for the unknown position of the body as a function
of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

An example of modelling a real world problem using differential
equations is the determination of the velocity of a ball falling through
the air, considering only gravity and air resistance. The ball's
acceleration towards the ground is the acceleration due to gravity minus
the deceleration due to air resistance. Gravity is considered constant,
and air resistance may be modeled as proportional to the ball's
velocity. This means that the ball's acceleration, which is a derivative
of its velocity, depends on the velocity. Finding the velocity as a
function of time involves solving a differential equation.

Differential equations are mathematically studied from several
different perspectives, mostly concerned with their solutions —the set
of functions that satisfy the equation. Only the simplest differential
equations admit solutions given by explicit formulas; however, some
properties of solutions of a given differential equation may be
determined without finding their exact form. If a self-contained formula
for the solution is not available, the solution may be numerically
approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.